System and method for x-ray absorption spectroscopy using a crystal analyzer and a plurality of detector elements

ABSTRACT

A fluorescence mode x-ray absorption spectroscopy apparatus includes an electron bombardment source of x-rays, a crystal analyzer, the source and the crystal analyzer defining a Rowland circle having a Rowland circle radius (R), a detector, and at least one stage configured to position a sample at a focal point of the Rowland circle with the detector facing the sample.

CLAIM OF PRIORITY

This application is a continuation from U.S. patent application Ser. No.17/320,852 filed May 14, 2021, which claims the benefit of priority toU.S. Provisional Appl. No. 63/026,613 filed on May 18, 2020, each ofwhich is incorporated in its entirety by reference herein.

BACKGROUND Field

The present application relates generally to x-ray absorptionspectroscopy systems.

Description of the Related Art

X-ray absorption spectroscopy (XAS) is a widely used technique fordetermining the local atomic geometric and/or electronic states ofmatter. XAS data is typically obtained by tuning the photon energy,often using a crystalline monochromator, to a range where core electronscan be excited (1-30 keV). The edges are, in part, named by which coreelectron is excited: the principal quantum numbers n=1, 2, and 3,correspond to the K-, L-, and M-edges, respectively. For instance,excitation of a 1s electron occurs at the K-edge, while excitation of a2s or 2p electron occurs at an L-edge.

XAS measures the x-ray absorption response of an element in a materialmatrix over an energy range across one of the absorption edge(s) of theelement, including the K-, and M-edges. respectively. There are threemain spectral regions in a XAS spectrum: 1) The pre-edge spectral regionbefore the peak absorption energy (white line); 2) The X-ray AbsorptionNear-Edge Structure (XANES) region, also called NI XAFS (Near-edge X-rayAbsorption Fine Structure) in the energy range from about 10 eV up toabout 150 eV above the white line and 3) EXAFS (Extended X-rayAbsorption Fine Structure) region in the energy range up to 1000 eVabove and including the absorption edge.

Transmission mode XAS measures x-rays transmitted through an objectcontaining the element of interest. XAS spectra are measured withsufficiently high x-ray energy resolution, (e.g., ranging from 0.3 eV to10 eV), depending on the spectral region of a XAS spectrum and theenergy of the absorption edge. For an x-ray source emitting x-rays overa wide energy bandwidth, a single crystal analyzer is typically used toselect a narrow energy bandwidth according to Bragg's law:

$\begin{matrix}{{2{d \cdot \sin}\;\theta} = {n\;\lambda}} & (1)\end{matrix}$

where d is the lattice spacing of the crystal analyzer, θ is the Braggangle, n is an integer, and λ is the wavelength of x-rays that satisfiesBragg's law. X-rays of wavelengths equal to λ/n diffracted by higherMiller index crystal planes of a crystal analyzer are referred to ashigh order harmonics. Additionally, lower Miller index crystal planeswith larger d-spacing reflect x-rays with proportionally largewavelength(s), referred to as low order harmonics.

SUMMARY

In certain implementations described herein, an apparatus comprises anx-ray source comprising a target configured to generate x-rays uponbombardment by electrons. The apparatus further comprises a crystalanalyzer positioned relative to the x-ray source on a Rowland circle ina tangential plane and having a Rowland circle radius (R). The crystalanalyzer comprises crystal planes curved along at least one directionwithin at least the tangential plane with a radius of curvaturesubstantially equal to twice the Rowland circle radius (2R). The crystalplanes are configured to receive x-rays from the x-ray source and todisperse the received x-rays according to Bragg's law. The apparatusfurther comprises a spatially resolving detector configured to receiveat least a portion of the dispersed x-rays. The spatially resolvingdetector comprises a plurality of x-ray detection elements having atunable first x-ray energy and/or a tunable second x-ray energy. Theplurality of x-ray detection elements are configured to measure receiveddispersed x-rays having x-ray energies below the first x-ray energywhile suppressing measurements of the received dispersed x-rays abovethe first x-ray energy and/or to measure the received dispersed x-rayshaving x-ray energies above the second x-ray energy while suppressingmeasurements of the received dispersed x-rays below the second x-rayenergy. The first and second x-ray energies are tunable in a range of1.5 keV to 30 keV.

In certain implementations described herein, a fluorescence mode x-rayabsorption spectroscopy apparatus comprises a source of x-rays, acrystal, and a detector. The source and the crystal define a Rowlandcircle. The apparatus is configured to receive a sample at a focal pointof the Rowland circle with the detector facing a surface of the sample.

In certain implementations described herein, a method comprisescollecting an XANES spectrum. The method further comprises collecting anEXAFS spectrum having coarser resolution than does the XANES spectrum.The EXAFS spectrum overlaps the XANES spectrum in an energy region of atleast 30 eV. The method further comprises normalizing the XANES spectrumand the EXAFS spectrum to one another in the energy region and replacingthe EXAFS spectrum in the energy region with the XANES spectrum in theenergy region to generate a combined spectrum.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates two types of crystal analyzers: Johanncrystal analyzers (left) and Johansson crystal analyzers (right).

FIG. 2 is a plot of an example calculated energy broadening ΔE as afunction of Bragg angle θ_(B) for 8 keV x-rays with an x-ray source spotsize of 400 microns and a Rowland circle diameter (2R) of 500millimeters.

FIG. 3 is a plot of the x-ray line energy and the radiative line widthas a function of atomic number of elements within a sample beinganalyzed.

FIG. 4 schematically illustrates an example apparatus in accordance withcertain implementations described herein.

FIG. 5 schematically illustrates simulation ray tracings of dispersedx-rays downstream from an example cylindrically curved Johansson crystalanalyzer in accordance with certain implementations described herein.

FIG. 6 schematically illustrates simulation ray tracings of dispersedx-rays downstream from an example spherically curved Johansson crystalanalyzer in accordance with certain implementations described herein.

FIG. 7 schematically illustrates simulation ray tracings of dispersedx-rays downstream from an example spherically curved Johann crystalanalyzer in accordance with certain implementations described herein.

FIG. 8 shows simulated x-ray spectra from a tungsten (W) target, arhodium (Rh) target, and a molybdenum (Mo) target.

FIG. 9 schematically illustrates simulation ray tracings of dispersedx-rays downstream from an example cylindrically curved Johansson crystalanalyzer for various targets in accordance with certain implementationsdescribed herein.

FIG. 10 schematically illustrates an example apparatus configured forXAS measurements in accordance with certain implementations describedherein.

FIG. 11 schematically illustrates another example apparatus configuredfor XAS measurements in accordance with certain implementationsdescribed herein.

FIG. 12 schematically illustrates another example apparatus configuredfor XAS measurements in accordance with certain implementationsdescribed herein.

FIG. 13A schematically illustrates the tangential plane and the sagittalplane of an example apparatus configured to have the sample between thex-ray source and the crystal analyzer in accordance with certainimplementations described herein.

FIG. 13B schematically illustrates the tangential plane and the sagittalplane of an example apparatus configured to have the sample between thecrystal analyzer and the spatially resolving detector in accordance withcertain implementations described herein.

DETAILED DESCRIPTION Overview

There are several challenges for XAS systems that use a laboratory x-raysource for high quality and high throughput XAS measurement. Thesechallenges are largely rooted in the laboratory x-ray source andexisting XAS system designs. The challenges associated with the x-raysource include:

-   low x-ray source brightness that leads to long measurement times,-   large x-ray source spot size that necessitates crystal analyzers    operating at high Bragg angles, large source-to-crystal analyzer    distance, or combination thereof,-   presence of narrow characteristic spectral lines over an extended    x-ray energy range which are not useful for XAS measurement, and-   constraint on the maximum electron acceleration voltage to minimize    high order harmonics that could be reflected by the crystal    analyzer, reducing production of useful x-rays for XAS measurement.

The problems associated with crystal analyzer operating at high Braggangles include the use of high Miller index crystal planes of thecrystal analyzer with excessively narrow energy bandwidth, spectralcontamination due to high order harmonics reflected by higher Millerindex crystal planes of the crystal analyzer and/or low harmonics (nbeing a fractional integer) reflected by the lower Miller index crystalplanes of the crystal analyzer, limited x-ray tuning x-ray range percrystal analyzer, and large beam size change on the sample over thex-ray energy measured. The higher and/or lower harmonics of the x-rayenergy reduces the signal-to-noise ratio of XAS measurements and canlead to substantial reduction of the XAS measurement quality andthroughput. The problems also impose challenges to the x-ray detectorused to measure the transmitted x-rays in transmission mode XASmeasurement.

Additionally, several problems can hinder accurate x-ray spectrummeasurements that include 1) change of the relative position between thex-ray source, the crystal analyzer, and the detector during themeasurement; 2) angular stability of the crystal analyzer; and 3) changein the x-ray spectrum of the x-ray source during the measurement.

Certain implementations described herein advantageously circumvent atleast some of these problems to provide accurate x-ray spectrummeasurements. Additionally, certain implementations described hereinachieve higher data collection speed by collecting a large solid angleof x-rays emitted from the x-ray source.

In the 1970s and 1980s, numerous laboratory XAS systems were developedby academic groups in recognition of the capability of x-ray absorptionspectroscopy for materials analysis and convenience of laboratory basedXAS systems. Rigaku developed several models of commercial laboratoryXAS systems but abandoned them some time ago. Most laboratory XASsystems use an electron bombardment laboratory x-ray source, acylindrically bent crystal designed to operate in a Rowland circlegeometry, and a single element (e.g., point) x-ray detector (e.g., suchas ionization chamber or proportional counter). XAS spectra aretypically generated by scanning the x-ray energy point by point. Energyscans are achieved by rotating the bent crystal while moving both thecrystal and a detector along the Rowland circle.

Such conventional laboratory approaches suffered drawbacks that includeinsufficient energy resolution, poor XAS data quality, long measurementtimes, stringent sample preparation requirements associated with a largeilluminated area, requiring use of multiple crystals for acquisition ofa single spectrum, and manufacturing difficulties for standard operation(e.g., challenges associated with moving a heavy rotating anode x-raysource along the Rowland circle and/or moving the sample—which isdifficult when the sample of interest is placed in in situenvironments). Those drawbacks, in conjunction with increasingavailability of synchrotron x-rays source based XAS facilities, led tothe waning of laboratory based XAS developments.

FIG. 1 schematically illustrates two types of crystal analyzers: Johanncrystal analyzers (left) and Johansson crystal analyzers (right). Inearlier systems, the crystals were cylindrically bent: flat in onedimension and curved in the other like a portion of a cylinder.Johann-type and Johansson-type crystals differ in the broadening error.For a Johann-type crystal, the reflection points on the crystal awayfrom the center lie outside of the Rowland circle (see, FIG. 1), sox-rays reflected at these non-central points by the reticular planes ofthe crystal are focused at different points, causing focusingaberrations (e.g., Johann errors; also referred to as focusing error)arising from the geometry. These focusing aberrations lead todegradation of the energy resolution, which is dependent on the Braggangle OB and can be expressed by the following expression:

$\begin{matrix}{ɛ = {\frac{1}{2}{E\left( \frac{l}{4R} \right)}^{2}\cot^{2}\theta_{B}}} & (2)\end{matrix}$

where l is the crystal size (e.g., width) along the Rowland circle, andR is the Rowland circle diameter. This relation implies that thefocusing aberrations increase rapidly with lower Bragg angles andconsequently, high energy resolution measurements are to be performedusing high Bragg angles (usually greater than 70 degrees) where thecotangent of OB is small. In comparison, a Johansson crystal analyzerdoes not suffer from the Johann focusing aberrations because the crystalis ground such that its surface matches the Rowland circle and allpoints of the Johansson-type crystal are coincident on the Rowlandcircle (see, FIG. 1).

An example laboratory XAS system, developed by a group led by Prof.Jerry Seidler at the University of Washington, is based on aconventional laboratory x-ray source with recently developedcommercially available x-ray components, including a spherically bent(e.g., doubly curved, rather than the single curvature of cylindricallybent) Johann crystal analyzer and a silicon drift detector. Large Braggangles (e.g., greater than or equal to 55 degrees) are used to achieveminimal energy broadening of the x-rays reflected by the crystalanalyzer resulting from the Johann focusing error, the energy broadeningΔE given by:

$\begin{matrix}{{\Delta E} = {E\;\cot\theta_{B}\Delta w}} & (3)\end{matrix}$

where E is the x-ray energy and Δw is the angular width of the x-raysource as seen by the crystal. The use of high Bragg angles enables highenergy resolution (because source broadening from the finite source spotsize and Johann errors are minimized).

FIG. 2 is a plot of an example calculated energy broadening ΔE as afunction of Bragg angle θ_(B) for 8 keV x-rays with an x-ray source spotsize of 400 microns and a Rowland circle diameter (2R) of 500millimeters. The angular contribution (Δw) is the x-ray source spot sizedivided by the distance from the x-ray source to the crystal (which isequivalent to 2·R·sinθ_(B)). As shown in FIG. 2, high Bragg angles areused for large x-ray sources, with an energy resolution of 2 eV for 8keV x-ray being achieved at angles greater than or equal to 73 degrees.

At high Bragg angles, high Miller index diffraction planes ofhigh-quality single crystal materials (e.g., Si and Ge) are typicallyused, which leads to several important drawbacks. High Miller indexcrystal reflections have very narrow Darwin widths, which means that theenergy band pass of the crystal is substantially smaller than the x-rayenergy resolution desired for most XAS measurements (e.g., around 0.5 eVfor XANES at about 2 keV and 1 eV to 4 eV for higher energy absorptionedges of XANES). Therefore, high Miller index crystals act as anexcessively narrow energy filter which leads to a large loss of usefulsource x-rays and thus a waste of source x-rays. Furthermore,undesirable x-rays including high order harmonics can be reflected bythe crystal analyzer with higher Miller index crystal planes than lowerMiller index planes, resulting lower XAS spectrum quality.

Another major drawback of using high Bragg angles is the use of multiplecrystals. The energy change per degree of crystal rotation at high Braggangles is substantially smaller than at low Bragg angles, resulting inlimited energy coverage per crystal analyzer. Many crystal analyzers aretherefore utilized to cover an operation energy range. For example, fora Bragg angle of 30 degrees and 8 KeV x-rays, every degree of rotationof the crystal covers about 236 eV, whereas at 85 degrees every degreeof rotation of the crystal covers only about 12.4 eV. Hence, formeasurements of about 100 eV coverage, the use of lower Bragg angleseasily satisfies the covered range but not so at higher Bragg angles.Such a system with a wide energy coverage (e.g., 2 to 20 keV) that canaddress a large portion of the periodic table of elements would becumbersome and expensive by use of an impractically large number ofcrystals. Additionally, the beam size on a sample also changes withx-ray energy substantially faster at higher Bragg angles than at lowerBragg angles, leading to use of a homogeneous sample (e.g., good sampleuniformity) or lower XAS spectrum quality with a heterogenous sample.

A key challenge that existed in the 1970s and 1980s laboratory-basedx-ray systems was the presence of harmonics that contaminate the signal.Such earlier systems simply ran the x-ray source at an electronaccelerating voltage (e.g., kVp) that was less than the energy of thelowest higher order harmonic to avoid contamination of the spectra.Because the amount of bremsstrahlung x-rays produced by electronbombardment is proportional to the accelerating voltage, this reductionof accelerating voltage came at the expense of the x-ray sourceefficiency. To run the x-ray source at higher efficiency, Seidler'sgroup use silicon drift detectors (SDDs) to circumvent the problem ofharmonic contamination to the XAS spectrum, but new problems areintroduced. First, the dimension of the active area of the SDDs istypically quite small—around 4 to 12 mm in diameter, which can be muchsmaller than the beam reflected by the crystal analyzer along thedirection perpendicular to the Rowland circle plane and thus leads tolong data collection time. Use of SDDs also limits their system designto using a spherically bent Johann crystal analyzer (SBCA) because thesize of the detector utilizes point-to-point focusing. Due to Johannbroadening error, their system is limited to operation at high Braggangles. Another problem of SDDs is that the maximum count rate withacceptable linearity is less than 1,000,000 per second, limiting theiruse to systems with lower count rates.

Certain implementations disclosed herein provide a laboratory XAS system(e.g., apparatus) that circumvents at least some of the problems of theprevious laboratory XAS systems described above and that enablelaboratory XAS systems with exceptional capabilities.

In the description herein, the plane of the Rowland circle is referredto as the tangential plane, the direction along the tangential plane isreferred to as the tangential direction, and the direction perpendicularto the tangential plane is referred to as the sagittal direction.

The x-ray signal, given by the number of x-ray photons per unit time perunit energy bandwidth for a given sample of a certain thickness, can beapproximately described by:

$\begin{matrix}{{N\left( \frac{ph}{s} \right)}\mspace{31mu}\alpha\mspace{31mu} B*T*\Omega_{D}*\Omega_{S}*R*D*S_{D}*S_{S}*\Delta E*M} & (4)\end{matrix}$

where S_(D) and S_(S) are the x-ray generating spot size of the x-raysource (or the size of the detector aperture depending on the specificsystem design) in the tangential and sagittal directions, respectively,B is the brightness of the x-ray source (which depends on the x-raygenerating spot size of the x-ray source S_(D) and S_(S)), T is thex-ray transmission through a sample, Ω_(D) and Ω_(S) are the collectionangles of the crystal analyzer in the tangential plane and the sagittaldirection, respectively, R is the reflectivity of the crystal analyzer(which depends upon the Miller index of crystal reflection planes andthe choice of material used), D is the detection efficiency of thedetector, ΔE is the energy resolution of the system, and M is the numberof (energy) spectral modes measured at the same time, which is equal tothe energy range simultaneously measured divided by ΔE.

The energy resolution ΔE can be selected to meet a desired energyresolution for the XAS measurements. Several factors affect the energyresolution ΔE, including the geometrical broadening of the x-ray beam(which is determined by the x-ray source spot size and the distancebetween the x-ray source and the crystal analyzer), the Darwin width ofthe crystal, the penetration of the x-rays into the crystal, and thefinite size of the apertures, as expressed approximately by:

$\begin{matrix}{{\Delta\; E} = {E\;{{\cot(\theta)}\left\lbrack {\left( \frac{s_{S} + s_{D}}{4R\;\sin\;(\theta)} \right)^{2} + \left( \omega_{C} \right)^{2} + \left( \frac{2\ln\; 2{\cos(\theta)}}{\mu 4R} \right)^{2} + \mspace{410mu}\left( {\frac{\left( {s_{S} + s_{D}} \right)^{4}}{4\left( {16R} \right)^{4}}se{c^{2}(\theta)}cs{c^{2}(\theta)}} \right)^{2}} \right\rbrack}^{1/2}}} & (5)\end{matrix}$

where ω_(C) is the Darwin width of the crystal and μ is the linearabsorption coefficient of the sample for a particular x-ray energy. ForXAS measurement in the XANES region, the energy resolution ΔE can beless than the radiative line width inherent to the absorption edge dueto core hole broadening. For XAS measurement in the EXAFS spectralregion of energies higher than the XANES region, the energy resolutionΔE can be higher than the radiative line width and up to 10 eV.

Certain implementations described herein have an energy resolution ΔEthat, for a specific XAS measurement, obtains optimal trade-off betweenenergy resolution and measurement speed. If the energy resolution is toocoarse, the finer details of the XAS spectra are not obtained. However,if the energy resolution is too fine, a significant penalty is paid inthe form of throughput loss by the acquisition of spectra taking toolong, rendering the system impractical. Certain implementationsdescribed herein provide a judicious choice of x-ray source spot sizes,power loading, crystal choice, and the aperture openings that optimizethe tradeoff between throughput and energy resolution.

FIG. 3 is a plot of the x-ray line energy and the radiative line widthas a function of atomic number of elements within a sample beinganalyzed. As shown in FIG. 3, the radiative line width is typicallydependent on the energy of an x-ray absorption edge. In certainimplementations described herein, the XAS system is configured to havean energy resolution that matches the desired (e.g., required) energyresolution for the appropriate XAS measurements. For example, an energyresolution between 0.2× to 1× of the radiative line width can beselected for XAS measurements in the XANES spectral region based on thespecific applications. A coarser energy resolution can be selected forthe EXAFS region to efficiently use source x-rays to increasethroughput.

EXAMPLE IMPLEMENTATIONS

FIG. 4 schematically illustrates an example apparatus 100 in accordancewith certain implementations described herein. The apparatus 100comprises an x-ray source 110 comprising a target 112 configured togenerate x-rays 114 upon bombardment by electrons. The apparatus 100further comprises a crystal analyzer 120 positioned relative to thex-ray source 110 on a Rowland circle 150 in a tangential plane andhaving a Rowland circle radius (R). The crystal analyzer 120 comprisescrystal atomic planes 122 curved along at least one direction 124 with aradius of curvature substantially equal to twice the Rowland circleradius (2R). The crystal atomic planes 122 are configured to receivex-rays 114 from the x-ray source 110 and to disperse the received x-raysaccording to Bragg's law (e.g., the dispersed x-rays 126). The apparatus100 further comprises a spatially resolving detector 130 positioned at adistance (D) from a downstream side 128 of the crystal analyzer 120,with D less than or equal to 2R. The spatially resolving detector 130 isconfigured to receive at least a portion of the dispersed x-rays 126,the spatially resolving detector 130 comprising a plurality of x-raydetection elements 132 having a tunable first x-ray energy and/or atunable second x-ray energy. The plurality of x-ray detection elements132 are configured to measure received dispersed x-rays 126 having x-rayenergies below the first x-ray energy while suppressing measurements ofthe received dispersed x-rays 126 above the first x-ray energy and/or tomeasure the received dispersed x-rays 126 having x-ray energies abovethe second x-ray energy while suppressing measurements of the receiveddispersed x-rays 126 below the second x-ray energy. The first and secondx-ray energies are tunable in a range of 1.5 keV to 30 keV.

In certain implementations, the Rowland circle 150 is a circle tangentto a center of the surface of the crystal analyzer 120 impinged by thex-rays 114. The Rowland circle radius (R) in certain implementations isin a range of less than 100 millimeters, in a range of 100 millimetersto 200 millimeters, in a range of 200 millimeters to 300 millimeters, ina range of 300 millimeters to 500 millimeters, or in a range of 500millimeters to 1000 millimeters.

In certain implementations, the apparatus 100 further comprises at leastone stage 140 configured to position the crystal analyzer 120 withrespect to the x-ray source 110 on the Rowland circle 150 in thetangential plane with the curved direction 124 of the crystal atomicplanes 122 aligned to the tangential plane, and to position thespatially resolving detector 130 at the distance (D) from the downstreamside 128 of the crystal analyzer 120. For example, the at least onestage 140 can comprise at least one linear motion stage configured toadjust the position of the crystal analyzer 120 (e.g., alongsubstantially perpendicular x-, y-, and z-directions) and at least onerotational motion stage configured to adjust the orientation of thecrystal analyzer 120 (e.g., in substantially perpendicular pitch, yaw,roll angles about principal axes). For another example, the at least onestage 140 can comprise at least one linear motion stage configured toadjust the position of the spatially resolving detector 130 (e.g., alongsubstantially perpendicular x-, y-, and z-directions) and at least onerotational motion stage configured to adjust the orientation of thespatially resolving detector 130 (e.g., in substantially perpendicularpitch, yaw, roll angles about principal axes).

In certain implementations, the apparatus 100 further comprises a samplestage 160 configured to position a sample 162 for analysis eitherbetween the x-ray source 110 and the crystal analyzer 120 or between thecrystal analyzer 120 and the spatially resolving detector 130. Forexample, the sample stage 140 can comprise at least one linear motionsub-stage configured to adjust the position of the sample 162 (e.g.,along substantially perpendicular x-, y-, and z-directions) and at leastone rotational motion sub-stage configured to adjust the orientation ofthe sample 162 (e.g., in substantially perpendicular pitch, yaw, rollangles about principal axes).

In certain implementations, the crystal analyzer 120 is configured to beoperated at low Bragg angles (e.g., in the range of 10 degrees to 60degrees; in the range of 10 degrees to 40 degrees; in the range of 10degrees to 30 degrees). For example, the curved crystal atomic planes122 can comprise crystal atomic planes (e.g., atomic planes of a singlecrystal material selected from the group consisting of: silicon,germanium, and quartz) having low Miller indices (e.g., Si<111>;Si<220>; Ge<111>; Ge<400>) and that are bent (e.g., mechanicallydeformed to be curved) along the at least one direction 124 to have aradius of curvature (2R) in a range of 100 millimeters to 2000millimeters. In certain implementations, the curved, low Miller indexcrystal atomic planes 122 at low Bragg angles can advantageously relaxthe sample uniformity constraints because at low Bragg angles, thechange in the x-ray beam size on the sample 162 that contributes to theanalysis is small as the crystal analyzer 120 is rotated.

In certain implementations, by using such curved, low Miller indexcrystal atomic planes 122, the crystal analyzer 120 can have an energyresolution that is optimized according to a predetermined spectralregion of an x-ray absorption spectroscopy (XAS) measurement to be made.For example, the energy resolution can be selected to be from 0.2 to 1times a radiative line width of an element to be measured (see, e.g.,FIG. 3), the radiative line width due to core hole broadening for thepre-edge spectral region before the peak absorption edge energy and theX-ray Absorption Near-Edge Structure (XANES) region, and can be selectedto be from 1 to 5 times of the radiative line width for the EXAFSregion. Such energy optimization of the crystal analyzer 120 can enableefficient use of source x-rays 114 and fast data acquisition speeds.

In certain implementations, the curved, low Miller index crystal atomicplanes 122 provide a large energy change per degree of rotation of thecrystal analyzer 120, thereby enabling the crystal analyzer 120 to covera large energy range over a given rotation angular range for XASmeasurements. In certain such implementations, a small number of crystalanalyzers 120 can be used for XAS measurements over a large energyrange. For example, just two crystal analyzers 120, one with Ge<111>crystal atomic planes 122 and the other with Ge<200> crystal atomicplanes, operating with Bragg angles in a range of 10 degrees to 50degrees, can be sufficient for XAS measurements over an energy range of4 keV to 20 keV.

In certain implementations, the source x-ray collection angle (e.g.,efficiency) in the tangential plane (e.g., dispersion plane) for thecurved crystal atomic planes 122 can be larger than for flat crystalatomic planes, thereby producing a converging (e.g., focused) x-ray beamat the Rowland circle 150 in the tangential plane. For example, with anx-ray beam 126 focused on the Rowland circle 150, a single elementdetector can be used. In certain implementations, a slit aperture can beused on the Rowland circle 150 and at the upstream side of the detector130 to improve energy resolution. For example, for sizes of the crystalanalyzer 120 of the order of 50 millimeters to 100 millimeters canresult in a collection angle of about 0.1 radian to 0.3 radian of anarrow energy bandwidth in the tangential plane, which can be over twoorders of magnitude higher than with flat crystals for which theacceptance angle is determined by the Darwin width (e.g., in the rangeof 10 microradians to 50 microradians). In certain implementations, thecrystal atomic planes 122 can also be curved (e.g., bent) in thesagittal direction, thereby increasing the collection angle of thex-rays 126 in the sagittal direction as well.

In certain implementations, the apparatus 100 comprises a cylindricallycurved (e.g., bent) Johansson crystal analyzer 120, which can provide alarge x-ray collection angle in the tangential direction but a limitedx-ray collection angle in the sagittal direction for a given x-rayenergy. In certain other implementations, the apparatus 100 comprises aspherically curved (e.g., bent) Johansson crystal analyzer, aspherically curved (e.g., bent) Johann crystal analyzer, a cylindricallycurved (e.g., bent) Johann crystal analyzer, or an analyzer with Wittrygeometry.

FIG. 5 schematically illustrates simulation ray tracings of dispersedx-rays 126 downstream from an example cylindrically curved Johanssoncrystal analyzer 120 in accordance with certain implementationsdescribed herein. The example cylindrically curved Johansson crystalanalyzer 120 of FIG. 5 has a Rowland circle radius R equal to 300millimeters and a crystal plane bending radius 2R equal to 600millimeters, and the x-ray source 110 was simulated as a point sourceemitting x-rays 114 within an angle of ±40 milliradians in thetangential direction and ±48 milliradians in the sagittal direction, anda set of x-ray energies in a range of 8047 eV to 8054 eV, with each bandin FIG. 5 corresponding to a 1 eV energy band. The horizontal axis isthe distance in the tangential direction parallel to the Rowland circle150 and the vertical axis is the distance in the sagittal direction. Inthe leftmost plot, each of the bands corresponds to the surface area ofthe crystal analyzer 120 that reflects x-rays of one energy. The centralband corresponds to x-rays having energies of 8048 eV and the remainingbands correspond to x-rays with energies higher than 8048 eV. Each pairsof bands counted from the center stripe corresponds to an increase ofx-ray energy by 1 eV. The other plots of FIG. 5 show example x-rayspectral distributions of the dispersed x-rays 126 at various locationsrelative to the Rowland circle 150 (e.g., 200 millimeters inside theRowland circle 150, 50 millimeters inside the Rowland circle 150, 5millimeters inside the Rowland circle 150, and on the Rowland circle150).

As seen in FIG. 5, the example cylindrically curved Johansson crystalanalyzer 120 can disperse a wide energy range in the sagittal directionwith high energy x-ray dispersion. The dispersed x-rays 126 can bemeasured by the detector 130 with at least some detection elements 132positioned along the sagittal direction to differentiate the angularlydispersed x-rays 126 along the sagittal direction (e.g., a detector 130with multiple detector elements 132 along the sagittal direction withsufficient spatial resolution can resolve the spectrum). In this way,the apparatus 100 can measure a spectrum over a finite x-ray energyrange with sufficiently high energy resolution. Furthermore, the spatialresolution for measuring the spectrum can be significantly relaxed whenthe detector 130 is positioned close to the crystal analyzer 120 (e.g.,within the Rowland circle 150).

In certain implementations in which some or all of the detectionelements 132 have at least one energy threshold (e.g., the tunable firstand second x-ray energies) to define an XAS energy bandwidth of interest(e.g., 50 eV to 5 keV), the signal-to-noise ratio of the XAS spectrumcan be improved by suppressing measurement of (e.g., rejecting) x-rays126 either above the XAS energy bandwidth of interest, therebysuppressing (e.g., rejecting) one or more harmonics diffracted by thecrystal analyzer 120 and/or by rejecting x-rays 126 with x-ray energiesbelow the XAS energy bandwidth of interest by suppressing (e.g.,rejecting) fluorescence x-rays.

In certain implementations, the apparatus 100 comprises a sphericallycurved (e.g., bent) Johansson crystal analyzer 120, which can provide alarge x-ray collection angle in the tangential direction and higherx-ray collection angle in the sagittal direction for a given x-rayenergy resolution than can the cylindrically curved Johansson crystalanalyzer 120. The spherically curved Johansson crystal analyzer 120 candisperse an x-ray energy range in both the tangential direction and thesagittal direction and the dispersed x-rays 126 can be measured with adetector 130 with a plurality of detection elements 132 configured tomeasure the angularly dispersed x-rays 126 along the tangentialdirection, achieving a spectrum over a finite x-ray energy range withhigh energy resolution.

FIG. 6 schematically illustrates simulation ray tracings of dispersedx-rays 126 downstream from an example spherically curved Johanssoncrystal analyzer 120 in accordance with certain implementationsdescribed herein. The example spherically curved Johansson crystalanalyzer 120 of FIG. 6 has a Rowland circle radius R equal to 300millimeters and a crystal plane bending radius 2R equal to 600millimeters, and the x-ray source 110 was simulated as a point sourceemitting x-rays 114 within an angle of ±40 milliradians in thetangential direction and ±48 milliradians in the sagittal direction, anda set of x-ray energies in a range of 8045 eV to 8049 eV, with each bandin FIG. 6 corresponding to a 1 eV energy band. The horizontal axis isthe distance in the tangential direction parallel to the Rowland circle150 and the vertical axis is the distance in the sagittal direction. Inthe leftmost plot, each of the bands corresponds to the surface area ofthe crystal analyzer 120 that reflects x-rays of one energy. The centralband corresponds to x-rays having energies of 8048 eV and the remainingbands correspond to x-rays with energies less than 8048 eV. Each pairsof bands counted from the center stripe corresponds to decrease of x-rayenergy by 1 eV. The other plots of FIG. 6 show example x-ray spectraldistributions of the dispersed x-rays 126 at various locations relativeto the Rowland circle 150 (e.g., 200 millimeters inside the Rowlandcircle 150, 50 millimeters inside the Rowland circle 150, 5 millimetersinside the Rowland circle 150, and on the Rowland circle 150).

As seen in FIG. 6, the dispersed x-rays 126 can be measured by thedetector 130 with at least some detection elements 132 positioned alongthe sagittal direction to differentiate the angularly dispersed x-rays126 along the sagittal direction (e.g., a detector 130 with multipledetector elements 132 along the sagittal direction with sufficientspatial resolution can resolve the spectrum). In this way, the apparatus100 can measure a spectrum over a finite x-ray energy range withsufficiently high energy resolution. Furthermore, the spatial resolutionfor measuring the spectrum can be significantly relaxed when thedetector 130 is positioned close to the crystal analyzer 120 (e.g.,within the Rowland circle 150).

In certain implementations, the apparatus 100 comprises a sphericallycurved (e.g., bent) Johann crystal analyzer 120, which can provide alarge x-ray collection angle in the sagittal direction but limited x-raycollection angle in the tangential direction for a given x-ray energyresolution. The spherically curved Johann crystal analyzer 120 candisperse an x-ray energy range in the tangential direction and thedispersed x-rays 126 can be measured with a detector 130 with aplurality of detection elements 132 configured to measure the angularlydispersed x-rays 126 along the tangential direction, achieving aspectrum over a finite x-ray energy range with high energy resolution.

FIG. 7 schematically illustrates simulation ray tracings of dispersedx-rays 126 downstream from an example spherically curved Johann crystalanalyzer 120 in accordance with certain implementations describedherein. The example spherically curved Johann crystal analyzer 120 ofFIG. 7 has a Rowland circle radius R equal to 300 millimeters and acrystal plane bending radius 2R equal to 600 millimeters, and the x-raysource 110 was simulated as a point source emitting x-rays 114 within anangle of ±40 milliradians in the tangential direction and ±48milliradians in the sagittal direction, and a set of x-ray energies in arange of 8028 eV to 8048 eV, with each band in FIG. 7 corresponding to a4 eV energy band. The horizontal axis is the distance in the tangentialdirection parallel to the Rowland circle 150 and the vertical axis isthe distance in the sagittal direction. In the leftmost plot, each ofthe bands corresponds to the surface area of the crystal analyzer 120that reflects x-rays of one energy. The central band corresponds tox-rays having energies of 8048 eV and the remaining bands correspond tox-rays with energies less than 8048 eV. Each pairs of bands counted fromthe center stripe corresponds to decrease of x-ray energy by 4 eV. Theother plots of FIG. 7 show example x-ray spectral distributions of thedispersed x-rays 126 at various locations relative to the Rowland circle150 (e.g., 200 millimeters inside the Rowland circle 150, 50 millimetersinside the Rowland circle 150, 5 millimeters inside the Rowland circle150, and on the Rowland circle 150).

As seen in FIG. 7, the dispersed x-rays 126 can be measured by thedetector 130 with at least some detection elements 132 positioned alongthe sagittal direction to differentiate the angularly dispersed x-rays126 along the sagittal direction (e.g., a detector 130 with multipledetector elements 132 along the sagittal direction with sufficientspatial resolution can resolve the spectrum). In this way, the apparatus100 can measure a spectrum over a finite x-ray energy range withsufficiently high energy resolution. Furthermore, the spatial resolutionfor measuring the spectrum can be significantly relaxed when thedetector 130 is positioned close to the crystal analyzer 120 (e.g.,within the Rowland circle 150).

In certain implementations, the apparatus 100 comprises a toroidallycurved (e.g., bent) Johansson crystal analyzer 120, which like thespherically curved Johansson crystal analyzer 120, can provide a largex-ray collection angle in the tangential direction but higher x-raycollection angle than the cylindrically curved Johansson crystalanalyzer 120 in the sagittal direction for a given x-ray energyresolution. The toroidally curved Johansson crystal analyzer 120 candisperse the x-rays in both the tangential direction and the sagittaldirection, and the dispersed x-rays 126 can be measured with a detector130 with a plurality of detection elements 132 configured to measure theangularly dispersed x-rays 126 along the tangential direction, achievinga spectrum over a finite x-ray energy range with high energy resolution.

In certain implementations, the x-ray source 110 comprises a highefficiency (e.g., high brightness) x-ray source comprising an electronsource and at least one anode target 112 (e.g., having a size on theorder of microns) configured to generate x-rays upon being bombarded byelectrons from the electron source. The target 112 can be on or embeddedin a thermally conductive substrate (e.g., comprising diamond)configured to dissipate thermal energy from the target 112 that isgenerated by the electron bombardment of the target 112. Examples oftarget 112 materials include but are not limited to, Cu, Cr, Fe, Co, Ni,Zn, Al, Rh, Mo, Pd, Ag, Ta, Au, W, SiC, MgCl, or other metals ormetal-containing materials. Examples of x-ray sources 110 compatiblewith certain implementations described herein are disclosed by U.S. Pat.Nos. 10,658,145, 9,874,531, 9,823,203, 9,719,947, 9,594,036, 9,570,265,9,543,109, 9,449,781, 9,448,190, and 9,390,881, each of which isincorporated in its entirety by reference herein.

In certain implementations, the x-ray source 110 can comprise aplurality of targets 112 having different materials configured toprovide a continuous (e.g., smooth) x-ray energy spectrum over anextended x-ray energy range for XAS measurements. Certain suchimplementations can overcome a limitation of single target materialx-ray sources 110 in which the spectrum of the resultant x-rays 114 froma single target 112 includes characteristic lines over an extendedenergy range, and these characteristic lines are not suitable for XASmeasurements. For example, FIG. 8 shows simulated x-ray spectra from atungsten (W) target 112, a rhodium (Rh) target 112, and a molybdenum(Mo) target 112. The presence of the characteristic lines of the Wtarget 112 between 7.5 keV and 13 keV makes this spectral region fromthe W target 112 unsuitable for XAS measurements. In certainimplementations, an x-ray source 110 comprising a W target 112 and oneor both of a Rh target 112 and a Mo target 112 can provide a combinedcontinuous (e.g., smooth) energy spectrum over a range of 1.5 keV to 20keV by using the W target 112 for the ranges of 1.5 keV to 7.5 keV and13 keV to 20 keV and using the Rh target 112 and/or the Mo target 112for the range of 7.5 keV to 14 keV. While W is more efficient for x-rayproduction, which is proportional to the atomic number Z of the targetmaterial and the electron accelerating voltage, the characteristic linesin the x-ray spectrum from the W target 112 can contaminate the spectrumbetween 7.5 keV and 14 keV, so the Rh and/or Mo targets 112 can be usedinstead for this energy range.

In certain implementations, the size and shape of the target 112 areselected to optimize performance depending on the parameters andcharacteristics of the crystal analyzer 120. For example, for acylindrically curved Johansson crystal analyzer 120, the x-ray source110 can comprise a rectangular (e.g., line-shaped) target 112 having afirst dimension (e.g., width; in a range of 3 microns to 100 microns)that is substantially aligned along the tangential direction and asecond dimension (e.g., length; in a range of 10 microns to 4millimeters) that is substantially aligned along the sagittal direction.The ratio of the second dimension to the first dimension for obtaining agiven fractional energy resolution ΔE/E due to the size of the target112 can be approximately equal to (ΔE/E)^(−1/2)·cot(θ), where B is theBragg angle. For another example, for a spherically curved Johanncrystal analyzer 120 or for a spherically curved Johansson crystalanalyzer 120, the ratio of the second dimension to the first dimensionfor obtaining an given fractional energy resolution ΔE/E due to the sizeof the target 112 can be approximately equal to(ΔE/E)^(−1/2)·cot(θ)·sin(θ). The x-ray source spot size (e.g., the sizeof the electron beam spot bombarding the target 112) is the “apparent”source size when viewed at the take-off angle such that the electronfootprint on the target 112 is compressed along one axis (e.g., theapparent source spot size is one-tenth the size of the electronfootprint at a take-off angle of 6 degrees).

FIG. 9 schematically illustrates simulation ray tracings of dispersedx-rays 126 downstream from an example cylindrically curved Johanssoncrystal analyzer 120 for various targets 112 in accordance with certainimplementations described herein. The example cylindrically curvedJohansson crystal analyzer 120 of FIG. 9 has a Rowland circle radius Requal to 300 millimeters and a crystal plane bending radius 2R equal to600 millimeters, and the x-ray source 110 was simulated as emittingx-rays 114 within an angle of ±40 milliradians in the tangentialdirection and ±48 milliradians in the sagittal direction, and a set ofx-ray energies in a range of 8048 eV to 8054 eV, with each band in FIG.9 corresponding to a 1 eV energy band. For the targets 112 of FIG. 9,the first dimension substantially along the tangential direction is 20microns and one-half of the second dimension along the sagittaldirection is varied among 0 (e.g., point source), 0.2 millimeter, 1millimeter, and 2 millimeters. The horizontal axis is the distance inthe tangential direction parallel to the Rowland circle 150 and thevertical axis is the distance in the sagittal direction. In the topmostplot for the point source, each of the bands corresponds to the surfacearea of the crystal analyzer 120 that reflects x-rays of one energy. Thecentral band corresponds to x-rays having energies of 8048 eV and theremaining bands correspond to x-rays with energies higher than 8048 eV.Each pairs of bands counted from the center stripe corresponds to anincrease of x-ray energy by 1 eV. The other topmost plots for the othersizes of the x-ray source spot size of the target 112, the bands areless well-defined but are distinguishable even with the largestsimulated x-ray source spot size, indicating that energy resolution ofabout 1 eV can still be achieved. The other plots in FIG. 9 show thex-ray spectral distribution at two different locations relative to theRowland circle 150 (e.g., 5 millimeters inside the Rowland circle 150and on the Rowland circle 150).

As seen in FIG. 9, the second dimension of the target 112 substantiallyaligned with the sagittal direction can be much greater than the firstdimension of the target 112 substantially aligned with the tangentialdirection for the cylindrically curved Johansson crystal analyzer 120.The dispersed x-rays 126 can be measured by the detector 130 with atleast some detection elements 132 positioned along the sagittaldirection to differentiate the angularly dispersed x-rays 126 along thesagittal direction (e.g., a detector 130 with multiple detector elements132 along the sagittal direction with sufficient spatial resolution canresolve the spectrum). FIG. 9 shows that the x-ray source spot size ofthe target 112 along the sagittal direction can be increased to increasethe x-ray source power and to increase throughput while keeping theenergy resolution to a desired value (e.g., 1 eV). For example, a highresolution, two-dimensional detector 130 positioned at the Rowlandcircle 150 can be used as the energy bands are more clearly separated.For a detector 130 with a high spatial resolution, an x-ray source 110with an extended spot size and high power can be used.

In certain implementations, the x-ray source 110 can comprise at leastone grazing incidence mirror configured to substantially reflect thex-rays 114 from the target 112 in the sagittal plane. For example, thetarget 114 can emit a beam of x-rays 114 that has an angular range thatis larger than the acceptance of the crystal analyzer 120. A pair ofgrazing incidence plane mirrors can be placed (e.g., one at each side ofthe x-ray beam) to reflect portions of the x-ray beam that would missthe crystal analyzer 120. The at least one grazing incidence mirror canbe considered as at least one virtual source of x-rays positioned closeto the Rowland circle 150 and that can be used to monitor the backgroundsimultaneously with the central spectrum acquisition.

In certain implementations, the spatially resolving detector 130 isselected from the group consisting of: a pixel array photon countingdetector, a direct conversion charge coupled device (CCD) detector(e.g., configured to operate in single photon detection mode), acomplementary metal-oxide-semiconductor (CMOS) detector (e.g.,configured to operate in single photon detection mode), and a pluralityof silicon drift detectors (e.g., placed in close proximity to oneanother or integrated with one another). For example, the spatiallyresolving detector 130 can comprise a one-dimensional (1D) positionsensitive detector (e.g., strip detector) or a two-dimensional (2D)position sensitive detector. In certain implementations, atwo-dimensional (2D) spatially resolving detector 130 having sufficientspatial resolution can be used with a large x-ray source spot size toachieve sufficiently high energy resolution. While FIG. 4 schematicallyillustrates an example implementation in which the spatially resolvingdetector 130 is at least partially inside the Rowland circle 150, incertain other implementations, the spatially resolving detector 130 isat least partially on the Rowland circle 150 or at least partiallyoutside the Rowland circle 150.

In certain implementations, the detection elements 132 (e.g., pixels;complete detectors) of the spatially resolving detector 130 have spatialdimensions in at least one dimension configured to provide apredetermined energy resolution (e.g., in a range of 0.2 eV to 3 eV) anda predetermined energy range (e.g., in a range of 3 eV to 200 eV). Forexample, the detection elements 132 can each have a size along thetangential direction in a range of 3 microns to 200 microns and a sizealong the sagittal direction in a range of 3 microns to 5000 microns. Incertain implementations, the detection elements 132 are spatiallyseparated from one another such that each detection element 132corresponds to less than 3 eV (e.g., in a range of 0.5 eV to 3 eV) ofthe beam spread of the dispersed x-rays 126 at the detector 130 (e.g.,for XANES measurements) or can correspond to less than 10 eV (e.g., in arange of 1 eV to 10 eV) of the beam spread of the dispersed x-rays 126at the detector 130 (e.g., for EXAFS measurements). For example, toachieve high throughput, the apparatus 100 is configured to utilize anx-ray spectrum that has a breadth covered by the detector 130 whilesimultaneously having a number M of spectral modes (e.g., energy bands)measured at the same time by the detector 130, where M is given by thespectral coverage at the detector 130 divided by the energy bandwidthper detection element 132. For example, for a spectral coverage of 200eV at the detector 130 and an energy bandwidth of 2 eV per detectionelement 132, M is equal to 100. In certain implementations, thedetection elements 132 have a linearity at a count rate exceeding 10⁶photons per second (most silicon drift detectors are linear only to 0.5million photons per second).

In certain implementations, the spatially resolving detector 130 isconfigured to measure the distribution of the dispersed x-rays 126 fromthe crystal analyzer 120. For example, for a cylindrically curvedJohansson crystal analyzer 120, the detection elements 132 can beconfigured to measure angularly dispersed x-rays 126 along the sagittaldirection. For another example, for a spherically curved Johann crystalanalyzer 120, the detection elements 132 can be configured to measureangularly dispersed x-rays 126 along the tangential direction. Foranother example, for a spherically curved Johann crystal analyzer 120,at least some of the detection elements 132 can be configured to measureangularly dispersed x-rays 126 along the tangential direction. In thisexample, the size of each detector element 132 can be small, with thedetection element size in the tangential direction (D1) expressed by:2·R·sin(θ)·ΔE/E≥cot(θ)·D1 and the size along the sagittal direction (S2)of the x-ray source spot on the target 112 expressed by:2·R·sin(θ)·(ΔE/E)^(1/2)≥S2, where R is the Rowland circle radius and Bis the Bragg angle. In certain implementations, the center of thedetector 130 is positioned on the Rowland circle 150. The size along thesagittal direction (D2) of the detection elements 132 can be smallerthan S2.

In certain implementations, the detection elements 132 have at least onetunable energy threshold (e.g., selected by the user of the apparatus100 or automatically by a computer-based controller) to suppress (e.g.,reject) x-rays outside the energy range of interest and that degrade theXAS measurements. For example, higher order harmonics can generate abackground signal contribution of 10% in the measured spectrum, whichcan reduce the throughput by about 3×. In certain implementations, thedetection elements 132 are configured to measure x-rays 126 having x-rayenergies below a tunable first x-ray energy while suppressingmeasurements of x-rays 126 above the tunable first x-ray energy. In thisway, the detection elements 132 can suppress (e.g., reject) higher orderharmonics diffracted by the crystal analyzer 120, thereby improving thequality of the measured XAS spectrum. Another benefit of suppressingmeasurements of higher energy x-rays is that the x-ray source 110 can beoperated at higher accelerating voltages for higher x-ray flux andthroughput. For another example, low energy x-rays (e.g., reflectedx-rays from the crystal analyzer 120; fluorescence x-rays from thesample 162) that satisfy Bragg's law can be received by the detector 130and can contribute to the background signal. In certain implementations,the detection elements 132 are configured to measure x-rays 126 havingx-ray energies above a tunable second x-ray energy while suppressingmeasurements of x-rays 126 below the tunable second x-ray energy. Inthis way, the detection elements 132 can suppress (e.g., reject) the lowenergy x-rays contributing to the background signal and to provide highquality XAS spectra. In certain implementations, the detection elements132 have both a tunable first x-ray energy and a tunable second x-rayenergy.

In certain implementations, the detector 130 comprises an aperturebetween the crystal analyzer 120 and the detection elements 132. Forexample, the aperture can be configured to have an adjustable size(e.g., by a user of the apparatus 100; by a computer-based controller)to controllably adjust an energy resolution of the detector 130. Foranother example, the aperture can be structured (e.g., have a pattern ofopenings) such that the shape of the x-rays 126 received by the detector130 ensure centering of components on the Rowland circle 150. In certainimplementations, in which the detection elements 132 have largedimensions and the size in the tangential direction of the x-ray sourcespot on the target 112 is large, a small aperture in front of thedetection elements 132 can be used to achieve small detector dimensions.

In certain implementations, the apparatus 100 is used to measuretransmission mode XAS spectra in an improved manner as compared toconventional XAS systems. For example, using a conventional XAS system,a complete transmission mode XAS measurement can be performed byscanning the angle of the crystal analyzer over an angular range tocover the x-ray energy range desired for the XAS measurement (e.g.,50-100 eV for XANES measurements; 300-1000 eV for EXAFS measurements).In contrast, in accordance with certain implementations describedherein, XAS measurements can be made by simultaneously collectingmultiple narrow spectra (e.g., finite energy ranges each narrower thanthe desired energy range for the XAS measurement) with energy rangesthat partially overlap with one another. These multiple spectra can benormalized and combined together appropriately to form the full XASmeasurement. In certain implementations, collecting the multiple spectrawith overlapping energy range can be used to minimize x-ray sourceintensity fluctuation between two spectra collected with overlappingenergy range.

For another example, the apparatus 100 can be used to provide EXAFSspectra with higher resolution than conventional EXAFS systems. Inaccordance with certain implementations described herein, an XASspectrum can be collected in the pre-edge region and in the XANESspectral region with energy resolution equal to or better than theradiative line width (e.g., by selecting the material of the crystalanalyzer 120 and the Miller indices of the crystal atomic planes 112 forhigher energy resolution) and in the full EXAFS spectral region (e.g.,in the spectral region away from the absorption edge) with coarserenergy resolution (e.g., 3-10 eV) but higher x-ray flux. The spectralregion of the full EXAFS by the spectrum can be replaced by the pre-edgeand XANES regions, with appropriate intensity normalization andstitching to generate a spectrum.

FIG. 10 schematically illustrates an example apparatus 100 configuredfor XAS measurements in accordance with certain implementationsdescribed herein. In certain such implementations, the apparatus 100comprises an x-ray source 110 comprising a target 112 having a smallx-ray source spot size in the tangential direction, a cylindricallycurved Johansson crystal analyzer 120, and a spatially resolvingdetector 130 with a plurality of detection elements 132 that arespatially resolving in at least one of the tangential direction and thesagittal direction. The target 112 of the x-ray source 110 can have alinear shape (e.g., a size along the tangential direction in a range of3 microns to 50 microns and a size along the sagittal direction in arange of 20 microns to 4000 microns), and the x-ray source 110 cancomprise multiple target materials. The cylindrically curved Johanssoncrystal analyzer 120 can comprise a single crystal of Si, Ge, or quartz,can be used at low Bragg angles (e.g., in a range of 15 degrees to 40degrees), can have a Rowland circle radius R in a range of 50millimeters to 1000 millimeters, and can have a crystal size less thanor equal to 100 millimeters in both the tangential direction and thesagittal direction. The spatially resolving detector 130 can beone-dimensional (1D) or two-dimensional (2D) and comprising at leastsome detection elements 132 along the sagittal direction. The detectionelements 132 can have at least one energy threshold to suppress (e.g.,reject) x-rays having energies above or below the XAS measurement x-rayenergy range.

FIG. 10 also schematically illustrates simulation ray tracings ofdispersed x-rays 126 downstream from a cylindrically curved Johanssoncrystal analyzer 120 for a spatially resolving detector 130 at twodifferent positions: (a) on the Rowland circle 150 for a point sourcetarget 112 and for an elliptical source target 112, and (b) 50millimeters inside the Rowland circle 150. The cylindrically curvedJohansson crystal analyzer 120 is that of FIG. 5 (having a Rowlandcircle radius R equal to 300 millimeters and a crystal plane bendingradius 2R equal to 600 millimeters), and a set of x-ray energies in arange of 8047 eV to 8054 eV, with each band in FIG. 10 corresponding toa 1 eV energy band and the central band corresponding to x-rays havingenergies of 8048 eV. The horizontal axis in the three plots of FIG. 10is the distance in the tangential direction parallel to the Rowlandcircle 150, the vertical axis is the distance in the sagittal direction,and the dimensions are in millimeters. As seen in FIG. 10, the x-rayenergy spread is along the sagittal direction, which is the longdimension of the spatially resolving detector 130. Sub-eV resolution isachieved by using detection elements 132 of sufficiently small size. Foran elliptical source, the spatially resolving detector 130 positioned atthe Rowland circle 150 can be two-dimensional (2D), and the spatiallyresolving detector 130 positioned within the Rowland circle 150 can beone-dimensional (1D).

In certain implementations, the apparatus 100 of FIG. 10 comprising thecylindrically curved Johansson crystal analyzer 120 is configured toachieve a given energy resolving power of (ΔE/E)⁻¹ by meeting thefollowing conditions: 2·R·sin(θ)·ΔE/E≥cot(θ)·S1 and2·R·sin(θ)·(ΔE/E)^(1/2)≥S2, where R is the Rowland circle radius, θ isthe Bragg angle, and S1 and S2 are the x-ray source spot sizes of thetarget 112 along the tangential direction and the sagittal direction,respectively. The width of the cylindrically curved Johansson crystalanalyzer 120 along the sagittal direction can be equal to N·S2, and thecylindrically curved Johansson crystal analyzer 120 can be configured toreceive and disperse x-rays 126 along the sagittal direction over anx-ray energy range approximately equal to N times of the energyresolution ΔE (e.g., N in a range of 2 to 100). The number of detectionelements 132 along the sagittal direction can be equal to or greaterthan N. The spatially resolving detector 130 can be positioneddownstream from the crystal analyzer 120 (e.g., at a distance D from adownstream side of the crystal analyzer 120, with the distance D in arange less than 2R (e.g., twice the Rowland circle radius R). For thespatially resolving detector 130 positioned at the Rowland circle 150,the size of the detection elements 132 in the tangential plane can becomparable (e.g., for better signal-to-noise ratio) or can be largerthan S1 (with potentially lower signal-to-noise ratios due to strayx-rays being measured). For detection elements 132 larger than S1, thespatially resolving detector 130 can comprise a slit aperture at anupstream side of the spatially resolving detector 130 (e.g., between thecrystal analyzer 120 and the detection elements 132) to suppress (e.g.,reject; reduce) the number of stray x-rays being measured.

FIG. 11 schematically illustrates another example apparatus 100configured for XAS measurements in accordance with certainimplementations described herein. In certain such implementations, theapparatus 100 comprises an x-ray source 110 comprising a target 112having a large x-ray spot size in the tangential direction, acylindrically curved Johansson crystal analyzer 120, and a spatiallyresolving detector 130 with a plurality of detection elements 132 (e.g.,2 to 2000) that are spatially resolving in at least the sagittaldirection. The target 112 of the x-ray source 110 can have a linearshape (e.g., a size along the tangential direction in a range of 3microns to 50 microns and a size along the sagittal direction in a rangeof 20 microns to 4000 microns), and the x-ray source 110 can comprisemultiple target materials. The cylindrically curved Johansson crystalanalyzer 120 can comprise a single crystal of Si, Ge, or quartz, can beused at low Bragg angles (e.g., in a range of 15 degrees to 40 degrees),can have a Rowland circle radius R in a range of 50 millimeters to 1000millimeters, and can have a crystal size less than or equal to 100millimeters in both the tangential direction and the sagittal direction.The spatially resolving detector 130 can be one-dimensional (1D) ortwo-dimensional (2D) and comprising at least some detection elements 132along the sagittal direction. The detection elements 132 can have atleast one energy threshold to suppress (e.g., reject) x-rays havingenergies above or below the XAS measurement x-ray energy range.

In certain implementations, the apparatus 100 of FIG. 11 comprising thecylindrically curved Johansson crystal analyzer 120 is configured toachieve a given energy resolving power of (ΔE/E)⁻¹ by meeting thefollowing conditions: 2·R·sin(θ)ΔE/E≥cot(θ)·D1 and2·R·sin(θ)·(ΔE/E)^(1/2)≥S2, where R is the Rowland circle radius, B isthe Bragg angle, D1 is the size of the detection elements 132 along thetangential direction, and S2 is the size along the sagittal direction ofthe x-ray source spot on the target 112. The width of the cylindricallycurved Johansson crystal analyzer 120 along the sagittal direction canbe equal to N·S2, and the cylindrically curved Johansson crystalanalyzer 120 can be configured to receive and disperse x-rays 126 alongthe sagittal direction over an x-ray energy range approximately equal toN times of the energy resolution ΔE (e.g., Nin a range of 2 to 100). Thenumber of detection elements 132 along the sagittal direction can beequal to or greater than N. The spatially resolving detector 130 can bepositioned downstream from the crystal analyzer 120 (e.g., at a distanceD from a downstream side of the crystal analyzer 120, with the distanceD in a range less than 2R (e.g., twice the Rowland circle radius R). Forthe spatially resolving detector 130 with its center positioned at theRowland circle 150, the size D2 of the detection elements 132 along thesagittal direction can be smaller than S2. For detection elements 132with D1 larger than 2·R·sin(θ)·ΔE/E/cot(θ)·, the spatially resolvingdetector 130 can comprise a slit aperture at an upstream side of thespatially resolving detector 130 (e.g., between the crystal analyzer 120and the detection elements 132) and having a long dimension of the slitaperture aligned along the sagittal direction to suppress (e.g., reject;reduce) the number of stray x-rays being measured.

In certain implementations, the apparatus 100 comprises an x-ray source110 comprising a target 112 having a small x-ray spot size in thetangential direction, a spherically curved Johansson crystal analyzer120, and a spatially resolving detector 130 with a plurality ofdetection elements 132 (e.g., 2 to 2000) that are spatially resolving inat least the sagittal direction. The target 112 of the x-ray source 110can have a size along the tangential direction in a range of 3 micronsto 50 microns and a size along the sagittal direction in a range of 20microns to 1000 microns, and the x-ray source 110 can comprise multipletarget materials. The spherically curved Johansson crystal analyzer 120can comprise a single crystal of Si, Ge, or quartz, can have a Rowlandcircle radius R in a range of 50 millimeters to 1000 millimeters, andcan have a crystal size less than or equal to 100 millimeters in boththe tangential direction and the sagittal direction. The spatiallyresolving detector 130 can be one-dimensional (1D) or two-dimensional(2D) and comprising at least some detection elements 132 along thesagittal direction. The detection elements 132 can have at least oneenergy threshold to suppress (e.g., reject) x-rays having energies aboveor below the XAS measurement x-ray energy range.

In certain implementations, the apparatus 100 comprising the sphericallycurved Johansson crystal analyzer 120 is configured to achieve a givenenergy resolving power of (ΔE/E)⁻¹ by meeting the following conditions:2·R·sin(θ)·ΔE/E≤cot(θ)·S1 and 2·R·sin²(θ)·(ΔE/E)^(1/2)≥S2, where R isthe Rowland circle radius, θ is the Bragg angle, and S1 and S2 are thesizes of the x-ray source spot on the target 112 along the tangentialdirection and the sagittal direction, respectively. The spatiallyresolving detector 130 can be positioned downstream from the crystalanalyzer 120 (e.g., at a distance D from a downstream side of thecrystal analyzer 120, with the distance D in a range less than 2R (e.g.,twice the Rowland circle radius R). The detection elements 132 can beconfigured to measure dispersed x-rays 126 received by the spatiallyresolving detector 130 along the tangential direction. In certainimplementations, the spatially resolving detector 130 is aone-dimensional (1D) position sensitive detector positioned close to thecrystal analyzer 120 (e.g., within the Rowland circle 150) with thedetection elements 132 aligned along the tangential direction, while incertain other implementations, the spatially resolving detector 130 is atwo-dimensional (2D) position sensitive detector positioned close to theRowland circle 150 (e.g., on the Rowland circle 150).

FIG. 12 schematically illustrates another example apparatus 100configured for XAS measurements in accordance with certainimplementations described herein. In certain such implementations, theapparatus 100 comprises an x-ray source 110, a spherically curved Johanncrystal analyzer 120, and a spatially resolving detector 130 with aplurality of detection elements 132. FIG. 12 also schematicallyillustrates simulation ray tracings of dispersed x-rays 126 downstreamfrom the spherically curved Johann crystal analyzer 120 for a spatiallyresolving detector 130 at two different positions: (a) on the Rowlandcircle 150 and (b) 200 millimeters inside the Rowland circle 150. Thespherically curved Johann crystal analyzer 120 has a Rowland circleradius R equal to 300 millimeters and a crystal plane bending radius 2Requal to 600 millimeters. The set of x-ray energies of the bands in FIG.12 are in a range of 8028 eV to 8052 eV, with each band in FIG. 12corresponding to a 1 eV energy band and the central band correspondingto x-rays having energies of 8048 eV. For the spatially resolvingdetector 130 positioned on the Rowland circle 150, the dispersion of therange of x-ray energies is only over about 400 microns and spectralblurring can occur that degrades the energy resolution of the measuredXAS spectra. For the spatially resolving detector 130 positioned 200millimeters inside the Rowland circle, the corresponding x-ray energydispersion stretches over a few millimeters and can be resolved using apixelated detector. Energy dispersion is achieved in certainimplementations by placing the spatially resolving detector 130sufficiently far from the focal point on the Rowland circle 150 (e.g.,either inside the Rowland circle 150 or outside the Rowland circle 150).

In certain implementations, the x-ray energy illuminating a sample 162can vary depending on the location of the sample 162. FIG. 13Aschematically illustrates the tangential plane and the sagittal plane ofan example apparatus 100 configured to have the sample 162 between thex-ray source 110 and the crystal analyzer 120 in accordance with certainimplementations described herein. FIG. 13B schematically illustrates thetangential plane and the sagittal plane of an example apparatus 100configured to have the sample 162 between the crystal analyzer 120 andthe spatially resolving detector 130 in accordance with certainimplementations described herein. For the example apparatus 100 of FIGS.13A and 13B, the detection elements 132 of the spatially resolvingdetector 130 can measure x-ray absorption corresponding to a respectiveregion of the sample 162. In certain implementations, by scanning thex-ray energy over a sufficiently large range, a spectral image of thesample 162 can be obtained (e.g., x-ray imaging spectroscopy).

Although commonly used terms are used to describe the systems andmethods of certain implementations for ease of understanding, theseterms are used herein to have their broadest reasonable interpretations.Although various aspects of the disclosure are described with regard toillustrative examples and implementations, the disclosed examples andimplementations should not be construed as limiting. Conditionallanguage, such as “can,” “could,” “might,” or “may,” unless specificallystated otherwise, or otherwise understood within the context as used, isgenerally intended to convey that certain implementations include, whileother implementations do not include, certain features, elements, and/orsteps. Thus, such conditional language is not generally intended toimply that features, elements, and/or steps are in any way required forone or more implementations. In particular, the terms “comprises” and“comprising” should be interpreted as referring to elements, components,or steps in a non-exclusive manner, indicating that the referencedelements, components, or steps may be present, or utilized, or combinedwith other elements, components, or steps that are not expresslyreferenced.

Conjunctive language such as the phrase “at least one of X, Y, and Z,”unless specifically stated otherwise, is to be understood within thecontext used in general to convey that an item, term, etc. may be eitherX, Y, or Z. Thus, such conjunctive language is not generally intended toimply that certain implementations require the presence of at least oneof X, at least one of Y, and at least one of Z.

Language of degree, as used herein, such as the terms “approximately,”“about,” “generally,” and “substantially,” represent a value, amount, orcharacteristic close to the stated value, amount, or characteristic thatstill performs a desired function or achieves a desired result. Forexample, the terms “approximately,” “about,” “generally,” and“substantially” may refer to an amount that is within ±10% of, within±5% of, within ±2% of, within ±1% of, or within ±0.1% of the statedamount. As another example, the terms “generally parallel” and“substantially parallel” refer to a value, amount, or characteristicthat departs from exactly parallel by ±10 degrees, by ±5 degrees, by ±2degrees, by ±1 degree, or by ±0.1 degree, and the terms “generallyperpendicular” and “substantially perpendicular” refer to a value,amount, or characteristic that departs from exactly perpendicular by ±10degrees, by ±5 degrees, by ±2 degrees, by ±1 degree, or by ±0.1 degree.The ranges disclosed herein also encompass any and all overlap,sub-ranges, and combinations thereof. Language such as “up to,” “atleast,” “greater than,” “less than,” “between,” and the like includesthe number recited. As used herein, the meaning of “a,” “an,” and “said”includes plural reference unless the context clearly dictates otherwise.While the structures and/or methods are discussed herein in terms ofelements labeled by ordinal adjectives (e.g., first, second, etc.), theordinal adjectives are used merely as labels to distinguish one elementfrom another, and the ordinal adjectives are not used to denote an orderof these elements or of their use.

Various configurations have been described above. It is to beappreciated that the implementations disclosed herein are not mutuallyexclusive and may be combined with one another in various arrangements.Although this invention has been described with reference to thesespecific configurations, the descriptions are intended to beillustrative of the invention and are not intended to be limiting.Various modifications and applications may occur to those skilled in theart without departing from the true spirit and scope of the invention.Thus, for example, in any method or process disclosed herein, the actsor operations making up the method/process may be performed in anysuitable sequence and are not necessarily limited to any particulardisclosed sequence. Features or elements from various implementationsand examples discussed above may be combined with one another to producealternative configurations compatible with implementations disclosedherein. Various aspects and advantages of the implementations have beendescribed where appropriate. It is to be understood that not necessarilyall such aspects or advantages may be achieved in accordance with anyparticular implementation. Thus, for example, it should be recognizedthat the various implementations may be carried out in a manner thatachieves or optimizes one advantage or group of advantages as taughtherein without necessarily achieving other aspects or advantages as maybe taught or suggested herein.

What is claimed is:
 1. A fluorescence mode x-ray absorption spectroscopyapparatus comprising: a source of x-rays comprising at least one targetconfigured to generate x-rays upon bombardment by electrons; a crystalanalyzer, the source and the crystal analyzer defining a Rowland circlehaving a Rowland circle radius (R); a detector; and at least one stageconfigured to position a sample at a focal point of the Rowland circlewith the detector facing the sample.
 2. The fluorescence mode x-rayabsorption spectroscopy apparatus of claim 1, wherein the crystalanalyzer is positioned on the Rowland circle in a tangential plane, thecrystal analyzer comprising crystal planes curved along at least onedirection within at least the tangential plane with a radius ofcurvature substantially equal to twice the Rowland circle radius (2R),the crystal planes configured to receive x-rays and to disperse thereceived x-rays according to Bragg's law, the detector comprising aspatially resolving detector configured to receive at least a portion ofthe dispersed x-rays.
 3. The fluorescence mode x-ray absorptionspectroscopy apparatus of claim 2, wherein the spatially resolvingdetector comprises a plurality of x-ray detection elements having atunable first x-ray energy and/or a tunable second x-ray energy, theplurality of x-ray detection elements configured to measure receiveddispersed x-rays having x-ray energies below the first x-ray energywhile suppressing measurements of the received dispersed x-rays abovethe first x-ray energy and/or to measure the received dispersed x-rayshaving x-ray energies above the second x-ray energy while suppressingmeasurements of the received dispersed x-rays below the second x-rayenergy, the first and second x-ray energies tunable in a range of 1.5keV to 30 keV.
 4. The fluorescence mode x-ray absorption spectroscopyapparatus of claim 3, wherein the first and second x-ray energies differfrom one another by an energy in a range of 50 eV to 5 keV.
 5. Thefluorescence mode x-ray absorption spectroscopy apparatus of claim 3,wherein at least three x-ray detection elements of the plurality ofx-ray detection elements are configured to measure at least somedispersed x-rays along a direction perpendicular to the tangential planeand/or at least three x-ray detection elements of the plurality of x-raydetection elements are configured to measure at least some dispersedx-rays along a direction parallel to the tangential plane.
 6. Thefluorescence mode x-ray absorption spectroscopy apparatus of claim 3,wherein the at least one stage is further configured to position thecrystal analyzer with respect to the source, to align the at least onedirection of the crystal planes to the tangential plane, and to positionthe spatially resolving detector at the distance D, wherein the at leastone stage is configured to rotate the crystal analyzer over apredetermined angular range while the relative positions of the source,the crystal analyzer, and the spatially resolving detector are changedto maintain Rowland circle geometry.
 7. The fluorescence mode x-rayabsorption spectroscopy apparatus of claim 3, wherein each x-raydetection element of the plurality of x-ray detection elements has alinear dimension between 3 microns to 2 millimeters.
 8. The fluorescencemode x-ray absorption spectroscopy apparatus of claim 3, wherein thedispersed x-rays have an energy bandwidth in a range of 2 eV to 250 eValong at least one direction perpendicular to or parallel to thetangential plane.
 9. The fluorescence mode x-ray absorption spectroscopyapparatus of claim 1, further comprising a sample stage configured toposition the sample for analysis either between the source and thecrystal analyzer or between the crystal analyzer and the detector. 10.The fluorescence mode x-ray absorption spectroscopy apparatus of claim1, wherein the detector is positioned inside the Rowland circle.
 11. Thefluorescence mode x-ray absorption spectroscopy apparatus of claim 1,wherein the detector is positioned on the Rowland circle.
 12. Thefluorescence mode x-ray absorption spectroscopy apparatus of claim 1,wherein the detector is positioned outside the Rowland circle.
 13. Thefluorescence mode x-ray absorption spectroscopy apparatus of claim 1,wherein the detector comprises a one-dimensional or two-dimensionalpixel array detector.
 14. The fluorescence mode x-ray absorptionspectroscopy apparatus of claim 13, wherein the pixel array detector hasan energy resolution better than 3 keV.
 15. The fluorescence mode x-rayabsorption spectroscopy apparatus of claim 1, wherein the crystalanalyzer is selected from a group consisting of: a cylindrically curvedJohansson crystal analyzer, a spherically curved Johansson crystalanalyzer, a spherically curved Johann crystal analyzer, a cylindricallycurved Johann crystal analyzer, and an analyzer with Wittry geometry.16. The fluorescence mode x-ray absorption spectroscopy apparatus ofclaim 1, wherein the Rowland circle radius is in a range of 50millimeters to 1000 millimeters.
 17. The fluorescence mode x-rayabsorption spectroscopy apparatus of claim 1, wherein the at least onetarget comprises one or more x-ray generating target materials selectedfrom the group consisting of: Cr, Fe, Co, Ni, Cu, Mo, Rh, Pd, Ag, Ta, W,and Au.
 18. The fluorescence mode x-ray absorption spectroscopyapparatus of claim 17, wherein the at least one target comprises adiamond substrate and the one or more x-ray generating target materialsare embedded on or within the diamond substrate.
 19. The fluorescencemode x-ray absorption spectroscopy apparatus of claim 1, wherein thesource has an effective source size less than 50 microns in a directionparallel to the tangential plane and/or has an effective source size ina range of 20 to 4000 microns in a direction perpendicular to thetangential plane.
 20. The fluorescence mode x-ray absorptionspectroscopy apparatus of claim 1, further comprising an aperturebetween the crystal analyzer and the spatially resolving detector. 21.The fluorescence mode x-ray absorption spectroscopy apparatus of claim20, wherein the aperture is a slit aperture having a width in a range of3 microns to 1000 microns.
 22. The fluorescence mode x-ray absorptionspectroscopy apparatus of claim 1, further comprises a beam stopconfigured to prevent the x-rays from the source that are not diffractedby the crystal analyzer from being received by the detector.
 23. Thefluorescence mode x-ray absorption spectroscopy apparatus of claim 1,further comprising a computer system configured to analyze signalsgenerated by the detector, to respond to the signals by generating anx-ray spectrum of x-ray flux as a function of x-ray energy, and toconvert x-ray spectra obtained with different x-ray energies to a singlecombined x-ray spectrum.
 24. The fluorescence mode x-ray absorptionspectroscopy apparatus of claim 1, further comprising a computer systemconfigured to record x-ray absorption spectra measured by the detectorand to produce a spectroscopic image of the sample under analysis.